Original EASE-Grid Format Description

EASE-Grid: A Versatile Set of Equal-Area Projections and Grids

Authors: Mary J. Brodzik and Ken Knowles, NSIDC

A version of this paper was presented at the NCGIA International
Conference on Discrete Global Grids, March 28, 2000, Santa Barbara,
California USA. The proceedings of that meeting were subsequently published by
NCGIA. The version of this paper published by NCGIA is available
at: http://www.ncgia.ucsb.edu/globalgrids-book/ease_grid/

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Citation

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The Equal-Area Scalable Earth Grid (EASE-Grid) consists of a set of three
equal-area projections, combined with an infinite number of possible grid
definitions. It is based on a philosophy of digital mapping and gridding
definitions that was developed at the National Snow and Ice Data
Center, in Boulder, Colorado USA. This philosophy was used to implement a library of
software routines, which are based on the assumption that a gridded data set is
completely defined by two abstractions, the map projection and an overlaid
lattice of grid points. The complete source code is available by following the Mapx link on the Data Analysis and Imaging Tools Web page, and contains
software to convert among many projections, but this document is intended as an
overview of the family of specific projections and grids that we have called the
NSIDC EASE-Grid, or simply EASE-Grid.

The EASE-Grid is intended to be a versatile tool for users of global-scale
gridded data, specifically remotely sensed data, although it is gaining
popularity as a common gridding scheme for data from other sources as well. Data
from various sources can be expressed as digital arrays of varying grid
resolutions, which are defined in relation to one of the three possible
projections. The user will find that visualization and intercomparison
operations are then greatly simplified, and that the tasks of analysis and
intercomparison can be more readily accomplished.

EASE-Grid Map Parameters

The three EASE-Grid projections comprise two azimuthal equal-area
projections, for the Northern or Southern hemisphere, respectively, and a
global cylindrical equal-area projection. All projections are based on a spherical model of the Earth
with radius R = 6371.228 km. This radius defines a sphere with the same surface area as the 1924 International Ellipsoid (also known as the International 1924 Authalic Sphere).

Northern Hemisphere Azimuthal Equal-Area Projection

The North azimuthal equal-area map is defined by the following equations:

r = 2*R/C * sin(lambda) * sin(PI/4 - phi/2) + r0

s = 2*R/C * cos(lambda) * sin(PI/4 - phi/2) + s0

h = cos(PI/4 - phi/2)

k = sec(PI/4 - phi/2)

Where:

Table 1. Definitions of Variables in Projection Equations

Variable

Definition

r

Column coordinate

s

Row coordinate

h

Particular scale along meridians

k

Particular scale along parallels

lambda

Longitude in radians

phi

Latitude in radians

R

Radius of the Earth = 6371.228 km

C

Nominal cell size

r0

Map origin column

s0

Map origin row

Note: The values of C, r0, and s0 are determined by the grid that
is chosen to overlay the projection. See the EASE-Grid Family of Grid Definitions section of this document for details.

Why Equal-Area Maps?

Discussion of map projections is often unnecessarily lengthy and
sidetracked by ignorance or disregard for the fact that there is no one
best map projection. Each projection has different properties and thus
different best uses. Sometimes the question is raised as to why we
chose equal-area projections over the other possibilities for the
EASE-Grids, and the answer relies on a basic understanding of projection
characteristics.

"Two of the most important characteristics of maps are whether they are conformal or equal-area. No map projection is both, and some are neither" (Knowles, 1993). On equal-area maps, a small circle placed anywhere on the map will always cover the same amount of area on the globe, and, at any point on the map, the product of the scale h along a meridian of longitude and the scale k along a parallel of latitude is always one. The aspect ratio k:h is a measure of shape distortion.

For the Northern and Southern hemisphere EASE-Grids, the aspect ratio
varies from 1:1 at the pole to 1.17:1 at 45N and increases to only 2:1 at
the equator. For the global EASE-Grid, the aspect ratio varies more
widely (see details in the following table). The selection of +/-30 for
the standard parallels of the cylindrical projection gives a map with
minimum mean angular distortion over the continents. This projection is
intended for the study of parameters in the mid- to low-latitudes.

In contrast, on conformal maps, angles within a small area are reproduced
accurately, so a small circle on the globe will look like a small circle
on the map. At any point on the map, the scale h along a meridian of
longitude is equal to the scale k along a parallel of latitude, and hk - 1
is a measure of areal distortion.

For example, the Polar Stereographic map true at 70N that is used for the
SMMR and SSM/I polar gridded data distributed by NSIDC is a conformal map.
By definition, the aspect ratio remains 1:1 everywhere; however, the areal
distortion of this map varies from -6 percent at the pole to +29 percent at 45° N and
increases to +276 percent at the equator.

Areal distortion of the Polar Stereographic map true at 70° N:

Polar Stereographic, (true at 70° N)

latitude

kh - 1

90

-6%

45

29%

0

276%

A very popular map that is neither equal-area nor conformal is the
cylindrical equidistant map, also known as the lat-lon grid. This map
suffers from both areal and shape distortion, as follows:

Shape Distortion

Areal Distortion

latitude

k/h

kh - 1

89

57

5630%

80

6

476%

60

2

100%

45

1.4

41%

0

1

0%

In summary, given the choices of either shape distortion or areal
distortion or both, we decided in favor of the equal-area projections for
the EASE-Grids because they minimized the amount of distortion over the
hemispheric and global scale we were attempting to portray. One
convenient side effect of this choice is that calculations of areal
statistics are reduced to simply summing pixels and multiplying by a
constant area per pixel, so the acronym, EASE, takes on a secondary meaning, as in easy to use. Users wishing a more general discussion of projection characteristics should also read Points, Pixels, Grids, and Cells -- A Mapping and Gridding Primer document.

Why a Spherical Earth Model?

Another question that is sometimes raised is why we chose to use a
spherical earth model over an elliptical model, and how much error this introduces in the gridding geolocation. The answer is that no error is introduced by this model choice.

Representation of the gridded data as a fixed array of values is
accomplished with a set of equations to map from geographic coordinates
(latitude, longitude) to grid coordinates (column, row). In this sense,
the location (column and row) of each grid "cell" can just be considered an entry in a look-up table -- a place to store the data (brightness temperature, albedo, time stamp, etc.) for a specific, implicitly defined, geographic location. As long as the transformation back from grid coordinates (column,row) to geographic coordinates (latitude, longitude) is performed with the inverse transformation that uses the same Earth model, there is no error introduced by using a spherical Earth model. Choice of an elliptical model would only slow down the transformation calculations (geographic to grid and back) with no gain in accuracy.

The fastest calculations, of course, would simply involve mapping to the cylindrical equidistant projection that was mentioned in the previous
section, since, in that projection, the latitude and longitude values
are, in effect, the column and row coordinates. However, that projection choice was rejected for reasons of unacceptable distortion in the output gridded data. Please see the previous section discussion, Why Equal-Area Maps?, for more information.

EASE-Grid Family of Grid Definitions

A grid is always defined in relation to a specific map projection. It is
essentially the parameters necessary to define a transparent piece of
graph paper that is overlaid on a flat map and then anchored to it at the
map origin. The following four elements completely describe a grid:

the map projection

the numbers of columns and rows

the number of grid cells per map unit (the map unit is part of the projection parameters)

the grid cell that corresponds to the map's origin

Any number of grid definitions can therefore be used to describe the
effect of changing the "graph paper" (for example, using fewer columns or rows,
a higher resolution, anchoring the map origin to the center of the grid,
etc.).

An array of gridded data, then, consists of one data element for each
grid cell or lattice point. The user has complete flexibility to
define the meaning of each grid cell value, according to the most
appropriate binning technique for the data and application at hand.

The EASE-Grid family of grid definitions includes, but is not limited to,
the following specific grids:

The Original SSM/I Grids

The original 25 kilometer grids were defined for the data products
generated by the SSM/I Level 3 Pathfinder Project at NSIDC, which includes gridded Passive Microwave Brightness Temperatures and a set of geophysical
products derived from the Brightness Temperatures. However, subsets of the grids for the azimuthal projections have been adopted by a number of other projects, including the TOVS and AVHRR Polar Pathfinders, and the
AARI (Arctic and Antarctic Research Institute, St. Petersburg, Russia) Sea
Ice data that have been regridded to EASE-Grid by NSIDC.

These grids have a nominal cell size of 25 km x 25 km. A slightly larger
actual cell size C=25.067525 km was chosen to make the full global, 25 km grid (ML) exactly span the equator and was then used for all three projections for the sake of data product consistency. Of course, few cells actually have these dimensions, but they all have the same area.

By convention, grid coordinates (r,s) start in the upper left corner, at
cell (0,0), with r increasing to the right and s increasing
downward. Rounding the grid coordinates up at .5 yields the grid cell
number. Grid cell is centered at grid coordinates (j,i) and bounded
by: j -.5 <= R < J +.5, I -.5 <= S < I +.5.

The 25 km hemispheric grids for the North and South azimuthal projections
(NL and SL, respectively) are defined with 721 columns, 721 rows, and the
respective pole anchored at cell (360.0,360.0). The ML grid for
the cylindrical projection is defined with 1383 columns, 586 rows, and is defined with the point where the equator crosses the prime meridian at cell location (691.0,292.5).

For each 25 km grid, the set of corresponding 12.5 km grids was defined such that the grid coordinates are coincident (bore-centered) and exactly double the lower resolution grid coordinates. The ML grid is symmetrical about the prime meridian, but the MH grid is not. The 25 km ML grid exactly spans the equator, from 180 W to 180 E, with 1383 grid cells. The global 12.5 km grid (MH) also exactly spans the equator, with 2766 grid cells. However, since the center of the ML column 0 is coincident with the ML column 0, the western edge of the MH grid cell in column 0 row 293 (at the equator) is slightly east of 180° W, and the eastern edge of the MH grid cell in column 2765 is slightly east of 180° E.

The dimensions, center, and extent of the original SSM/I grids are
summarized below. It is important to remember that there is nothing
specific to the SSM/I data in these definitions. If these grid
definitions are considered appropriate for another data set, they can
be used with no changes.

Original 25 km and 12.5 km Grids

Grid Name

Projection/
Resolution

Dimensions

Map Origin

Map Origin

Grid Extent

Width

Height

Column (r0)

Row (s0)

Latitude

Longitude

Minimum Latitude

Maximum Latitude

Minimum Longitude

Maximum Longitude

ML

Global
25 km

1383

586

691.0

292.5

0.0

0.0

86.72S

86.72N

180.00W

180.00E

MH

Global
12.5 km

2766

1171

1382.0

585.0

0.0

0.0

85.95S

85.95N

179.93W

180.07E

NL

Northern Hemisphere
25 km

721

721

360.0

360.0

90.0N

0.0

0.34S

90.00N

180.00W

180.00E

NH

Northern Hemisphere
12.5 km

1441

1441

720.0

720.0

90.0N

0.0

0.26S

90.00N

180.00W

180.00E

SL

Southern Hemipshere
25 km

721

721

360.0

360.0

90.0S

0.0

90.00S

0.34N

180.00W

180.00E

SH

Southern Hemisphere
12.5 km

1441

1441

720.0

720.0

90.0S

0.0

90.00S

0.26N

180.00W

180.00E

Other Grid Definitions in the EASE-Grid Family

The Polar Pathfinders

Users of the NSIDC EASE-Grid are not limited to the grid orientation, size
and resolution described above, and are free to define grids that are more
appropriate to a given data set. For example, the TOVS Polar Pathfinder
data were defined with the EASE-Grid Northern hemisphere map projection
parameters, and a polar subset of the original hemisphere at a 100
kilometer resolution. The AVHRR Polar Pathfinder data were defined for
both Northern and Southern hemisphere maps, as subsets of each, at 1.25
km, 5 km, and 25 km resolutions. The figure below shows the grid extent
for SSM/I, TOVS Polar, and AVHRR Polar grids.

The AARI Sea Ice Data in EASE-Grid

The AARI EASE-Grid sea ice data provide another example. These data did not require hemispheric coverage, but the data set producers at NSIDC wanted to provide them in a grid that would facilitate intercomparison with sea ice data derived from SSM/I. Therefore the AARI EASE-Grid was defined to be the subset of the SSM/I Pathfinder NH grid (Northern hemisphere, 12.5 km resolution) defined by columns 360 through 1080 and rows 360 through 1080. The resulting AARI EASE-Grid is 721 columns and 721 rows. This, in turn, relates the AARI EASE-Grid definition to the 25 km AVHRR EASE-grid (aka "NA25") subset via the following simple relationship:

AARIcolumn = 2 * NA25column

AARIrow = 2 * NA25row

For example, the center of the (12.5 km) AARI grid cell at
(column,row)=(0,0) corresponds to the center of the (25 km) NA25 grid cell
(0,0). The AARI grid cell at (1,0) corresponds to the NA25 grid cell (0.5,0),
etc. Since these grids were based on the original SSM/I grids, which
were defined to be bore-centered, the extent of the AARI grid is therefore
one half of one 12.5 km cell inward from the extent of the NA25 grid displayed
in the image above. Here is an example of the grid extent boundaries
at the upper left corner of the AARI and NA25 EASE-Grids.